Angle Bisector Theorem

Drag points B or C. Notice how D moves to keep the angles equal.

Angles (Bisector Check)

∠ BAD 0°
∠ CAD 0°
(Equal? Yes)

Side Lengths

AB 0
AC 0
BD 0
DC 0

The Ratios

AB AC
= 0.000
BD DC
= 0.000

Interactive Proof

1. Analyze the Angles

Look at the data panel above. Notice that ∠BAD = ∠CAD. Let's call this angle x.

Let ∠ADB = θ. Because angles on a straight line sum to 180°, ∠ADC = 180° - θ.

2. Sine Formula on Left Triangle (ΔABD)

Applying the sine rule:

BD sin(x)
=
AB sin(θ)

Rearranging for the ratio of sides:

sin(x) sin(θ)
=
BD AB
... (Equation 1)

3. Sine Formula on Right Triangle (ΔACD)

Applying the sine rule:

DC sin(x)
=
AC sin(180° - θ)

Critical Trig Fact: sin(180° - θ) is exactly the same as sin(θ)!

So:
sin(x) sin(θ)
=
DC AC
... (Equation 2)

4. Conclusion

Both Equation 1 and Equation 2 equal

sin(x)sin(θ)
.

Therefore:

BD AB
=
DC AC

Rearranging terms gives the standard form:

AB AC
=
BD DC